Linearizability of the perturbed Burgers equation

Linearizability of the perturbed Burgers equation

R. A. Kraenkel, J. G. Pereira, and E. C. de Rey Neto

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil ~Received 7 August 1997; revised manuscript received 18 November 1997!

We show in this report that the perturbed Burgers equation ut52uux1uxx1e(3a1u 2

ux13a2uuxx 13a3ux

2

1a4uxxx) is equivalent, through a near-identity transformation and up to O(e), to a linearizable equation if the condition 3a123a32

3 2a21

3

2a450 is satisfied. In the case this condition is not fulfilled, a

normal form for the equation under consideration is given. We show, furthermore, that nonlinearizable cases lead to perturbative expansions with secular-type behavior. Then, to illustrate our results, we make a lineariz-ability analysis of the equations governing the dynamics of a one-dimensional gas.@S1063-651X~98!01908-4# PACS number~s!: 03.40.Kf

I. INTRODUCTION

The object of this paper is the perturbed Burgers equation

ut52uux1uxx1e~3a1u2ux13a2uuxx13a3ux 2_1a

4uxxx!,

~1!

whereaiare constants,e!1 is a perturbative parameter, and

subscripts denote partial differentiation. It appears in the long-wave, small-amplitude limit of extended systems domi-nated by dissipation, but where dispersion is also present at a higher order. More precisely, those systems described by equations whose linear part admits a dispersion relation of the form

V~k!5a3k31a5k51•••1ı~b2k21b4k41•••!, ~2!

with aiand bireal constants. For example, Eq.~1! appears in

the description of gas dynamics @1#, and in certain cases of free-surface motion of waves in heated fluids @2#. More im-portant, however, is the fact that the terms appearing at order

e are the only ones allowed if Eq.~1! is obtained from long-wave perturbation theory, and no constants are allowed to scale withe. In this sense, it has the universality character-istics, much in the same way as the equations discussed by Calogero@3#.

When the O(e) terms are discarded, we have simply a Burgers equation, which is an equation linearizable through a Hopf-Cole transformation @4#. It is, thus, a natural question to know when Eq.~1! is also linearizable. Put in this way, the answer is that it is linearizable ifa15a25a35a4, in which

case the equation is reduced to the sum of Burgers with the first higher-order equation of the Burgers hierarchy@5#. We notice in passing that the latter is also linearizable by the same Hopf-Cole transformation that linearizes the Burgers equation. However, we can put the question on a more gen-eral setting by introducing the idea of near identity

transfor-mation@6#, that is, a transformation u→w of the form

u5w1ef~w!. ~3!

If we apply such a transformation to Eq. ~1!, we may look for functionsf(w) such that the transformed equation reads:

w_t52ww_x1w_xx1el~3w2w_x13ww_xx

13wx

2_1w

xxx!1O~e2!, ~4!

for some lPR. If such a f(w) exists, we say that Eq.~1! and Eq.~4! are equivalent up to O(e), and this is the equiva-lence that is physically relevant, as long as perturbation theory is concerned. As Eq.~4! is linearizable, so is Eq. ~1! up to O(e). The fundamental issue here is thus to determine the conditions for the existence of a near identity transfor-mation@that is,f(w)# ensuring the equivalence, up to O(e), of Eqs. ~1! and ~4!. This is the question we will address in this paper, and an answer will be given in terms of a condi-tion on the parameters a1,a2,a3, and a4.

This kind of equivalence is usually introduced in the con-text of normal form analysis of ordinary differential equa-tions, and has been discussed for dispersive partial differen-tial equations in @7,8#. When the lowest order, in the long-wavelength limit, of such system is described by the Korteweg–de Vries equation ~KdV!, there exists always a near-identity transformation that makes the O(e) perturba-tions integrable. Indeed, in Ref.@8#, it has been shown that, by introducing a f(w) depending explicitly on x, one can completely remove the O(e) corrections. However, obstacles to integrability appear at O(e2) @7,9,10#. For a similar analy-sis in the case of the nonlinear Schro¨dinger equation, the reader is addressed to@11#.

We will show that, in the case of Eq. ~1!, obstacles to linearizability appear already at O(e). We mean by this that, in general, Eq.~1! is not equivalent to Eq. ~4!. The condition for the equivalence will be shown to be

3a123a32

3 2a21

3

2a450. ~5!

Furthermore, in the case where condition~5! is not satisfied, we find a normal form for Eq.~1!. Finally, as an illustration, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas, and we show that, already at order O(e), these equations cannot be linearized.

II. LINEARIZABILITY ANALYSIS

Let us then implement the ideas exposed above. We want to insert Eq. ~3! into Eq. ~1!, discard all O(e2) terms, and compare the result with Eq.~4!. To do so, we have to specify PRE 58

the possible form of f(w). They ought to be such as to generate, at O(e), terms of the form w2wx, wwxx, wx

2

, and

wxxx. The allowable terms turn out to be wx, w2, and wx]21w, where]21means integration in x. Thus the general

form off(w) is

f~w!5awx1bw21gwx]21w, ~6!

wherea,b, and g are constants to be determined. We introduce now the following useful notations:

B~u!52uux1uxx, ~7!

and

Q~u!53a1u2ux13a2uuxx13a3ux 2_1a

4uxxx. ~8!

This makes Eq. ~1! read

ut5B~u!1eQ~u!. ~9!

The transformation ~3! changes Eq. ~9! to an equation in w, given by

wt5B~w!1e$Q~w!1@B~w!,f~w!#%1O~e2!, ~10!

where

@B~w!,f~w!#5d_dB_wf2df_d

wB.

In order to obtain the transformed equation, we have thus to calculate the commutator@B(w),f(w)#, which is the tedious part of our task. After performing that calculation, we get

@B~w!,f~w!#52bw_x212gww_xx1~2b1g!w2w_x.

~11!

Inserting this into Eq. ~10!, the transformed equation reads

wt5B~w!1e@~2b1g13a1!w2wx1~2g13a2!wwxx

1~2b13a3!wx 2_1a

4wxxx#. ~12!

If we now require Eq. ~12! to be of the form given by Eq.

~4!, we have to take l5a4, and the following conditions

must be satisfied:

2b53a423a3, ~13!

2g53a423a2, ~14!

2b1g53a423a1. ~15!

Clearly, this system of equations is not always solvable. The solubility condition is 3a123a32 3 2a21 3 2a450, ~16! in which caseb53

2(a42a3) andg532(a42a2). Note that a is left undetermined. Condition~16! is thus the condition that must be satisfied in order to make Eq.~1! equivalent, up to O(e), to Eq.~4!.

Suppose now that Eq. ~16! is not satisfied. The general form of the transformed equation is given by Eq.~12!. The

O(e) terms can be written as the sum of a linearizable term proportional toa4F3(w), with

F₃~w!53w2_w

x13wwxx13wx

2_1w

xxx, ~17!

plus a term Z(w) representing the obstacle to linearizability, that is, w_t5B~w!1ea₄F₃~w!1eZ~w!, ~18! where Z~w!5~2b13a323a4!wx 2 1~2g13a223a4!wwxx 1~2b1g13a123a4!w2wx. ~19!

If we call each of the coefficients appearing in the obstacle respectively by m₁,m₂, and m₃, and if we further introduce

ni throughmi5mni, withm53a123a32

3 2a21

3

2a4, then

we may write out the normal form of Eq.~1! as

w_t5B~w!1ea₄F₃~w!1em~n₁w_x21n₂ww_xx1n₃w2_w x!,

~20!

wheren_iare arbitrary constants satisfying

n12n31 n2

2 521. ~21!

Equation~20! encompasses the main results of this letter: for

m50 we have a linearizable equation, and formÞ0 it gives

the general form to which Eq.~1! is equivalent up to O(e). III. HOPF-COLE TRANSFORMATION

AND PERTURBATIVE EXPANSIONS

A further simplification can be achieved if, instead of Eq.

~6!, we introduce, following Ref. @8#, a more general

trans-formation, given by

f~w!5awx1bw21gwx]21w1nx~wxx12wwx!,

~22!

withn a constant. This leads to the following normal form:

wt5B~w!1e~a412n!F3~w!1em~n1wx 2₁

n2wwxx

1n3w2wx!, ~23!

where m is not modified, relation ~21! is still valid, but the coefficientsn1 andn3 have new expressions in terms of the

parameters defining the transformations. Explicitly, we have

n15m21~2b13a323a422n!, ~24!

n35m21~2b1g13a123a422n!. ~25!

The meaning of this result is the following: the new trans-formation generated by Eq.~22! does not have any influence on the linearizability up to O(e) of Eq. ~1!, that is, it does not alter condition ~16!. But, it makes it possible to further simplify the normal form by taking n5a4/2. This

com-pletely eliminates the F₃(w) term from Eq.~23!. We are thus lead to the study of the equation

wt5B~w!1em~n1wx 2_1n

2wwxx1n3w2wx!. ~26!

Let us now proceed to make a Hopf-Cole transformation given by

w5fx

f . ~27!

Inserting in Eq. ~26! we get

fxtf2 fxft5 fxxxf2 fxfxx1ef2Z~ f !, ~28!

with Z( f )5m(n1wx 2₁

n2wwxx1n3w2wx). This means that,

in general, we do not have a linear equation anymore, as would be the case if Z( f )[0. Let us try to perform a per-turbative expansion for f by setting

f5 f01ef11•••. ~29!

We get, by comparing order by order in Eq.~28!,

L~ f0!50, ~30!

f0@L~ f1!#x2 f0xL~ f1!5 f0 2_{Z~ f}

0!, ~31!

where L(_•)5]t2]xx. As a consequence of these equations

we see that, if we have a solution of Eq.~30! such that there exist constants ni satisfying the constraint ~21!, for which Z( f0)50, then we may take f1[0. This means that f0 is an

exact solution to Eq.~28!. This is the case, for instance, for the shock-type solution

f0511eQ~k!, ~32!

withQ(k)5kx1k2t. If we taken152n252n352

2 3, then

Z( f₀)50. However, this is an exceptional case. To analyze more general situations, we first note that Eq. ~31! can be seen as a first-order linear nonhomogeneous equation for

L( f₁). Solving it, we get

L~ f1!5 f0

E

djZ~ f0!, ~33!

which is a diffusion equation with a space and time depen-dent source term. It is thus clear that, depending on the ‘‘in-tensity’’ of the source, Eq. ~33! may not have always bounded solutions. This is a signal that such perturbative expansions are not uniformly valid. This is not surprising, as such is also the case for the perturbed KdV equation@7#. It is useful to have a more explicit example. Let us take

f0511eQ~k1!1eQ~k2!. ~34!

It is not difficult to see that there do not exist ni allowing Z( f0)50. We take then the samenias in the shock solution

~32!. We come to the following expression for Z( f0):

Z~ f₀!52m

3 f₀2@k1k2~k12k2!

2_eQ~k₁!1Q~k₂!_{#. ~35!}

A general solution for Eq. ~33! is not possible in this case. Let us introduce a further approximation by putting k12k2

5Dk, and let us compute Z( f0) to the lowest order inDk.

Clearly, this gives

Z~ f0!5 2m~Dk!2k₁2 3

F

e2Q~k1! F2

G

1O~Dk 3_{!, ~36!}

where F5 f0uk₁5k₂5112eQ(k1). We can now explicitly

in-tegrate this expression, yielding the following equation for

f1:

L~ f₁!5~Dk!2k1m

6 @11F ln F#. ~37! Let us now look for a solution to Eq.~37! by supposing that

f15 f1„Q(k1)…. This implies the following ordinary

differen-tial equation for f1:

f₁

8

2 f₁

9

5~Dk! 2_m

6k1 @11F ln F#.

~38!

Instead of solving this equation directly, let us notice the following point. The variable that represents the first-order correction to w0 is]x( f1/ f0) @with w05( f0)x/ f0#.

Remem-bering that F5 f0uk_15k2 we are naturally led to define f15 Fg_„Q(k1)…. Inserting this definition in Eq. ~38!, we come to

an equation for g

8

that is exactly the relevant variable, which reads: g

9

~112eQ~k1!!1g

8

~122eQ~k1!! 5~Dk! 2_m 6k1 @11~112e Q~k1!!ln~112eQ~k1!!#. ~39!

The solution to this equation for g

8

can be found by elemen-tary methods, and is given by

g

8

~Q!52~Dk! 2_m 6k1 e2Q~k1!$1 41 1 2~112eQ~k1!! 3@11ln~112eQ~k1!!#%_. ~40! This solution diverges linearly with Q(k₁) for Q(k₁)→`, indicating that the perturbative expansion fails. This is quite similar to the secular behavior present in many cases of per-turbative expansions for differential equations @12#. We should notice, however, that this is not a usual secular term coming from linear resonances that are eliminated by proce-dures like the multiple-scale method. Finally, we note that the transformation given by Eq.~22! is not essential for the derivation of the above results, which could be obtained without making use of it.

IV. GAS DYNAMICS

Let us consider the equations governing the dynamics of a one-dimensional gas@1#

rt1~ru!x50, ~41!

wherer(x,t) is the density, u(x,t) is the velocity, m is the viscosity, and

P5A

S

r r0

D

g

is the pressure, with g5(cp/cv) the ratio of specific heats,

and A a proportionality constant. In order to study its long-wavelength, small-amplitude limit, we define slow space and time variables,

j5e~x2ct!, ~43!

t5e2_{t, ~44!}

and scale the original ~primed! density and velocity fields according to

r

8

5r01er, ~45!

u

8

5eu. ~46!

In terms of these new variables, Eqs.~41! and ~42! become

r0uj2crj1e@rt1~ur!j#50, ~47!

Ag

r0 rj2cr0

u_j1e@2c~ur!_j1r0ut1r0ut1r0~u2!j

2mu_jj#1e2@~ur!_t1~ru2!_j#50. ~48!

Moreover, as a compatibility condition at O(e0_{), we have to}

set

c25Ag r0

. ~49!

Now, from Eq. ~47! we obtain

rj5r_c0 uj1e_c@rt1~ur!j#1O~e2!, ~50! or r5r0 c

F

u1 e c~u 2_2]21~u t!!

G

1O~e2!, ~51! with]21 indicating an integration in thej coordinate. Sub-stituting in Eq. ~48!, and using the resulting equation into itself, we are led to

u_t52uu_j1 m 2r0 u_jj1e

F

1 cu 2_u j2 3m 2cr0 uu_jj 2_4cm_r 0~uj! 2₁ m 2 8cr₀2ujjj

G

1O~e 2_{!. ~52!}

In order to compare to Eq.~1!, we first have to rewrite Eq.

~52! in a nondimensional form. To this end, we

nondimen-sionalize all variables according to

u→u c, j→2 r0 cm j, t→ r0 2c2m t. ~53!

In terms of these new variables, the nondimensional version of Eq.~52! reads u_t52uu_j1u_jj1e

F

22r0 Agu 2_u j2 3r0 Ag uujj 2 r0 2Ag~uj! 2₂ r0 4Agujjj

G

1O~e 2_{!. ~54!}

A comparison with Eq.~1! yields

a152 2r0 3Ag, a252 r0 Ag, a352 r0 6Ag, a452 r0 4Ag. ~55!

The linearizability condition~16!, therefore, is 3r0

8Ag50. ~56!

This means that, in the long-wavelength, small-amplitude limit, the equation governing a one-dimensional gas can be linearized only at the lowest order. When the O(e) correc-tions are taken into account, the corresponding equation can-not be linearized, indicating that obstacles to linearizability are present already at this order.

V. FINAL REMARK

In the case of the traveling-wave solution to the Burgers equation, the above analysis simplifies considerably. As we have already shown, Z(w)50 for the shock solution. In this case, however, we could have looked directly to Eq. ~20! without need of a Hopf-Cole transformation. Indeed, if we take

w52k@12tanh~kx22k2_{t!#, ~57!}

which is a solution of the Burgers equation, we can promptly verify that n₁52n₂52n₃522

3 implies Z(w)50. This makes it easy to find the O(e) correction to the solution~57! coming from the joint solution satisfying simultaneously the Burgers and the first higher-order equation of the Burgers hierarchy, which can be verified to be @13#

w52k$12tanh@kx2~2k224ea4k3!t#%. ~58!

Finally, we note that, having obtained the function w(x,t), the physical variable u(x,t) is obtained through Eq. ~3!. In the case under examination, using the fact that a has been left arbitrary, we come to the following form for f(w), which represents the effects of the perturbation terms present in Eq.~1!:

f~x,t!522k2_{b tanh~Q¯!1gk}2_sech2_~Q¯!

3$ln@12~11e22Q

¯

!#%, ~59!

whereQ¯5kx2(2k224ea4k3)t. The first term represents a

modification of amplitude of the shock solution. The second term represents a deformation profile. Notice, however, that

this second term is important only in the region near the origin of the Q¯ axis, falling to zero as Q¯ increases. The relative importance of the deformation with respect to the amplitude modifications depends moreover on the coeffi-cients b andg, which in turn depend on the coefficients of Eq. ~1!.

Recently, we came to know that part of our results

con-cerning the existence of near-identity transformations had been independently obtained in Ref.@14#.

ACKNOWLEDGMENT

The authors would like to thank CNPq, Brazil, for partial financial support.

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